Statistics (university) · Part 02
Probability
Measure chance with rules for events, conditions and counting.
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- Sample space and eventsThe sample space is the list of every possible outcome. An event is a group of outcomes you care about. Probability is the number of outcomes in the event divided by the total, when all outcomes are equally likely.
- Addition ruleFor two events, P(A or B) = P(A) + P(B) - P(A and B). You subtract the overlap because it was counted twice. The complement rule says P(not A) = 1 - P(A).
- Mutually exclusive eventsTwo events are mutually exclusive if they cannot both happen at once. Their overlap is zero, so P(A or B) is just P(A) + P(B). This is different from being independent.
- Independent eventsEvents are independent if one happening does not change the chance of the other. Then P(A and B) = P(A) x P(B).
- Conditional probabilityConditional probability is the chance of A once you know B happened: P(A given B) = P(A and B) / P(B). Knowing B shrinks the sample space to B.
- Law of total probabilityIf events B1, B2, ... split all outcomes into separate cases, then P(A) is the sum of P(A given Bi) x P(Bi). You weigh the chance of A in each case by how likely the case is.
- Bayes' theoremBayes' theorem turns P(B given A) into P(A given B): P(A given B) = P(B given A) x P(A) / P(B). It updates a belief after seeing evidence, using the base rate.
- Permutations and combinationsA permutation counts arrangements where order matters: nPr = n! / (n - r)!. A combination counts selections where order does not matter: nCr = n! / (r! (n - r)!). The symbol n! means n x (n-1) x ... x 1.